When you see…

Find the zeros

You think…

To find the zeros...To find the zeros...

Set 0)( xf Factor or use quadratic formula if quadratic. Graph to find zeros on calculator.

When you see…

Find equation of the line tangent to f(x) at (a, b)

You think…

Equation of the tangent lineEquation of the tangent line

Take derivative of f(x) Set maf )( Use y- y1 = m ( x – x1 )

You think…

When you see…

Find equation of the line normal to f(x) at (a, b)

Equation of the normal lineEquation of the normal line

Take )(xf

Set )(

1

afm

Use y – y1 = m ( x – x1)

You think…

When you see…

Show that f(x) is even

Even functionEven function

f (-x) = f ( x)

y-axis symmetry

You think…

When you see…

Show that f(x) is odd

Odd functionOdd function

. f ( -x) = - f ( x )

origin symmetry

You think…

When you see…

Find the interval where f(x) is increasing

ff(x) increasing(x) increasing

Find 0)( xf Answer: ( a, b ) or a < x < b

You think…

When you see…

Find the interval where the slope of f (x) is increasing

Slope of Slope of f f (x) is increasing(x) is increasing

Find the derivative of )()( xfxf Set numerator and denominator = 0 to find critical points Make sign chart of )(xf

Determine where 0)( xf is positive

You think…

When you see…

Find the minimum value of a function

Minimum value of a functionMinimum value of a function

Find where 0)( xf To find critical numbers Plug those values and a and b

into f (x) Choose the smallest

You think…

When you see…

Find critical numbers

Find critical numbersFind critical numbers

Express )(xf as a fraction Set both numerator and denominator = 0

You think…

When you see…

Find inflection points

Find inflection pointsFind inflection points

Express )(xf as a fraction Set numerator and denominator = 0 Make a sign chart of )(xf Find where )(xf changes sign ( + to - ) or ( - to + )

You think…

When you see…

Show that exists lim ( )x a

f x

Show existsShow exists lim ( )x a

f x

Show that

xfxfaxax

limlim

You think…

When you see…

Show that f(x) is continuous

..f(x) is continuousf(x) is continuous

S h o w t h a t

1 ) xfax

lim e x i s t s ( p r e v i o u s s l i d e )

2 ) af e x i s t s

3 ) afxfax

lim

You think…

When you see…

Find vertical

asymptotes of f(x)

Find vertical asymptotes of f(x)Find vertical asymptotes of f(x)

Factor/cancel f(x), if possible

Set denominator = 0

You think…

When you see…

Find horizontal asymptotes of f(x)

Find horizontal asymptotes of f(x)Find horizontal asymptotes of f(x)

Show

xfx lim

and

xf

x lim

You think…

When you see…

Find the average rate of change of f(x) at [a, b]

Find the average value of the derivative of f(x) :

Average rate of change of f(x)Average rate of change of f(x)

ab

afbfdxxf

ab

b

a

)()()(

1

You think…

When you see…

Find the instantaneous rate of change of f(x)

At x=a

Instantaneous rate of change of f(x)Instantaneous rate of change of f(x)

Find )(af

You think…

When you see…

Find the average valueof xf on ba,

Average value of the functionAverage value of the function

Find

b-a

dxxfb

a

You think…

When you see…

Find the absolute maximum of f(x) on [a, b]

Find the absolute minimum of f(x)Find the absolute minimum of f(x)

a) Find the derivative of f(x) b) Find all critical numbers c) Plug those values into f (x) d) Find f (a) and f (b) e) Choose the largest of c) and d)

You think…

When you see…

Show that a piecewise function is differentiable at the point a where the function rule splits

Show a piecewise function is Show a piecewise function is

differentiable at x=adifferentiable at x=a

F i r s t , b e s u r e t h a t t h e f u n c t i o n i s c o n t i n u o u s a t

ax .

T a k e t h e d e r i v a t i v e o f e a c h p i e c e a n d s h o w t h a t

xfxfaxax

limlim

You think…

When you see…

Given s(t) (position function), find v(t)

Given position s(t), find v(t)Given position s(t), find v(t)

Find tstv

You think…

When you see…

Given v(t), find how far a particle travels (total distance traveled)on

[a, b]

Given v(t), find how far a particle Given v(t), find how far a particle travels on [a,b]travels on [a,b]

Find where the particle changes direction, that is where v(t) changes sign.

Find b

c

c

a

dttvdttv )()( where t = c

is where the particle changes direction.

Or on your calculator: dttv

b

a )(

You think…

When you see…

Find the average velocity of a particle

on [a, b]

Find the average rate of change on Find the average rate of change on [a,b][a,b]

Find

vta

b

dt

basbsaba

You think…

When you see…

Given v(t), determine if a particle is speeding up at

t = a

Given v(t), determine if the particle is Given v(t), determine if the particle is speeding up at t=aspeeding up at t=a

Find v (k) and a (k).

Multiply their signs.

If positive, the particle is speeding up.

If negative, the particle is slowing down

You think…

When you see…

Given v(t) and s(0),

find s(t)

Given v(t) and s(0), find s(t)Given v(t) and s(0), find s(t)

s t v t dt C

P lu g in t = 0 to fin d C

You think…

When you see…

Show that the Mean Value Theorem holds

on [a, b]

Show that the MVT holds on [a,b]Show that the MVT holds on [a,b]

S h o w th a t f is c o n tin u o u s a n d d if fe re n tia b leo n th e in te rv a l .

T h e n f in d s o m e c s u c h th a t

f c f b f a

b a.

You think…

When you see…

Find the domain

of f(x)

Find the domain of f(x)Find the domain of f(x)

Assume domain is

, .

Domain restrictions: non-zero denominators,Square root of non negative numbers,Log or ln of positive numbers

You think…

When you see…

Find the range

of f(x) on [a, b]

Find the range of f(x) on [a,b]Find the range of f(x) on [a,b]

Use max/min techniques to find relativemax/mins.

Then examine

f a , f b

You think…

When you see…

Find the range

of f(x) on ( , )

Use max/min techniques to find relativemax/mins.

Then examine

limx

f x .

Find the range of f(x) onFind the range of f(x) on ,

You think…

When you see…

Find f ’(x) by definition

Find f Find f ‘‘( x) by definition( x) by definition

f x limh0

f x h f x h

or

f x limx a

f x f a x a

You think…

When you see…

Rate of change of y is increasing

proportionally to y

.y is increasing proportionally to yy is increasing proportionally to y

kydt

dy

translating to

ktCey

You think…

When you see…

Find the line x = c that divides the area under

f(x) on [a, b] into two equal areas

Find the x=c so the area under f(x) is Find the x=c so the area under f(x) is

divided equallydivided equally

dxxfdxxfb

c

c

a

OR

dxxfdxxfb

a

c

a

2

1

You think…

When you see…

dttfdx

d x

a

Fundamental TheoremFundamental Theorem

2nd FTC: Answer is xf

You think…

When you see…

dtuf

dx

d u

a

Fundamental Theorem, againFundamental Theorem, again

2nd FTC: Answer is dx

duuf

You think…

When you see…

The rate of change of population is …

Rate of change of a population Rate of change of a population

...dt

dP

You think…

When you see…

The line y = mx + b is tangent to f(x) at (a, b)

.y = mx+b is tangent to f(x) at (a,b)y = mx+b is tangent to f(x) at (a,b)

Two relationships are true. The two functions share the same slope at (a, b) ( afm ) and share the same y value at (a, b).

You think…

When you see…

Find area using left Riemann sums

Area using left Riemann sumsArea using left Riemann sums

If subintervals have equal width:

)(...)()()( 1210

nxfxfxfxfn

abA

You think…

When you see…

Find area using right Riemann sums

Area using right Riemann sumsArea using right Riemann sums

If subintervals have equal width:

)(...)()()( 321 nxfxfxfxfn

abA

You think…

When you see…

Find area using midpoint rectangles

Area using midpoint rectanglesArea using midpoint rectangles

Typically done with a table of values.

Be sure to use only values that are given.

If you are given 6 sets of points, you can only do 3 midpoint rectangles.

You think…

When you see…

Find area using trapezoids

Area using trapezoidsArea using trapezoids

)()(2...)(2)(2)(2 1210 nn xfxfxfxfxf

n

abA

This formula only works when the width of each subinterval is the same.

If not, you have to do individual trapezoids

You think…

When you see…

Solve the differential equation …

Solve the differential equation...Solve the differential equation...

Separate the variables – x on one side, y on the other. The dx and dy must all be upstairs. Then integrate both sides, and solve for y if possible

You think…

When you see…

Meaning of

dttfx

a

Meaning of the integral of f(t) from a to xMeaning of the integral of f(t) from a to x

The accumulation function –

accumulated area under the function xf

starting at some constant a and ending at x

You think…

When you see…

Given a base, cross sections perpendicular to

the x-axis that are squares

Semi-circular cross sections Semi-circular cross sections perpendicular to the x-axisperpendicular to the x-axis

The area between the curves typically is the base of your square.

So the volume is dxbase

b

a 2

You think…

When you see…

Find where the tangent line to f(x) is horizontal

Horizontal tangent lineHorizontal tangent line

Write xf as a fraction.

Set the numerator equal to zero

You think…

When you see…

Find where the tangent line to f(x) is vertical

Vertical tangent line to f(x)Vertical tangent line to f(x)

Write xf as a fraction.

Set the denominator equal to zero.

You think…

When you see…

Find the minimum

acceleration given v(t)

Given v(t), find minimum accelerationGiven v(t), find minimum acceleration

First find the acceleration tvta

Then minimize the acceleration by examining ta .

You think…

When you see…

Approximate the value f(0.1) of by using the

tangent line to f at x = 0

Approximate f(0.1) using tangent line Approximate f(0.1) using tangent line to f(x) at x = 0to f(x) at x = 0

Find the equation of the tangent line to f using 11 xxmyy

where 0fm and the point is 0,0f .

Then plug in 0.1 into this line.Be sure to use an approximation sign.

You think…

When you see…

Given the value of F(a) and the fact that the

anti-derivative of f is F, find F(b)

Given Given FF((aa)) and the that the and the that the anti-derivative of anti-derivative of ff is is FF, find , find FF((bb))

Usually, this problem contains an antiderivative you cannot take. Utilize the fact that if xF is the antiderivative of f,

then aFbFdxxfb

a

.

Solve for bF using the calculator to find the definite integral

You think…

When you see…

Find the derivative off(g(x))

Find the derivative of f(g(x))Find the derivative of f(g(x))

xgxgf

You think…

When you see…

Given , find dxxfb

a

dxkxfb

a

Given area under a curve and vertical Given area under a curve and vertical shift, find the new area under the curveshift, find the new area under the curve

dxkdxxfdxkxfb

a

b

a

b

a

You think…

When you see…

Given a graph of

find where f(x) is

increasing

'( )f x

Given a graph of f ‘(x) , find where f(x) is Given a graph of f ‘(x) , find where f(x) is increasingincreasing

Make a sign chart of xf

Determine where xf is positive

You think…

When you see…

Given a chart of x and f(x) on selected values between a and b, estimate where c is between a and b.

'( )f x

Straddle c, using a value k greater than c and a value h less than c.

so hk

hfkfcf

You think…

When you see…

Given , draw a

slope field dx

dy

Draw a slope field of dy/dxDraw a slope field of dy/dx

Use the given points

Plug them into dx

dy,

drawing little lines with theindicated slopes at the points.

You think…

When you see…

Find the area between curves f(x) and g(x) on

[a,b]

Area between f(x) and g(x) on [a,b]Area between f(x) and g(x) on [a,b]

dxxgxfAb

a ,

assuming f (x) > g(x)

You think…

When you see…

Find the volume if the area between the curves

f(x) and g(x) is rotated about the x-axis

Volume generated by rotating area between Volume generated by rotating area between f(x) and g(x) about the x-axisf(x) and g(x) about the x-axis

dxxgxfAb

a 22

assuming f (x) > g(x).

You think…

When you see…

Find the minimum slope of a function

Minimum slope of a functionMinimum slope of a function

Make a sign chart of f ’(x) = f ” (x) Find all the relative minimums Plug those back into f ‘ (x ) Choose the smallest

You think…

When you see…

Find the derivative of the inverse of f(x) at x = a

Derivative of the inverse of f(x) at x=aDerivative of the inverse of f(x) at x=a

Interchange x with y.

Solve for dx

dy implicitly (in terms of y).

Plug your x value into the inverse relation and solve for y.

Finally, plug that y into your dx

dy.

You think…

When you see…

Given v(t) and s(0), find the greatest distance from the origin of a particle on [a, b]

Given Given vv((tt)) and and ss(0)(0), find the greatest distance from , find the greatest distance from the origin of a particle on [the origin of a particle on [aa, , bb]]

Generate a sign chart of tv to find turning points. Integrate tv using 0s to find the constant to find ts . Find s(all turning points) which will give you the distance from your starting point.

When you see…

Given a water tank with g gallons initially being filled at the rate of F(t) gallons/min and emptied at the rate of E(t) gallons/min on

, find

1 2[ , ]t t

You think…

a) the amount of water in

the tank at m minutes

Amount of water in the tank at t minutesAmount of water in the tank at t minutes

dttEtFgt

t 2

You think…

b) the rate the water

amount is changing

at m

Rate the amount of water is changing at t = m

mEmFdttEtFdt

d m

t

You think…

c) the time when the

water is at a minimum

The time when the water is at a minimumThe time when the water is at a minimum

mEmF = 0,

testing the endpoints as well.

When you see…

Find a 3rd degree Taylor polynomial centered at a for f(x)

You think…

33rdrd degree Taylor polynomial... degree Taylor polynomial...

!3

))((

!2

))(())(()()(

32

3

axafaxafaxafafxP

Never include a term that has a power of x greater than the degree (3 in this case). If you use your polynomial to approximate a value of the function, don’t use an equal sign.

When you see…

You think…

][arcsin xdx

d

Derivative of arcsin:Derivative of arcsin:

21

1

x

You think…

When you see…

Find the radius of convergence of a power series centered at x=a

Radius of convergence:Radius of convergence:

Use the ratio test: 1lim 1

n

n

n b

b

Write result as: rax r is the radius of convergence

You think…

When you see…

Find the interval of convergence of a power

series centered at a.

Interval of convergence:Interval of convergence:

Find the radius of convergence (previous slide) Add and subtract r from a to create a

preliminary interval Substitute each endpoint into the power series

(for x) and determine if the resulting series converges or diverges.

If the series converges include the endpoint in the interval. If it diverges, don’t include the endpoint.

You think…

When you see…

Find the error bound of an alternating,

decreasing series.

Error boundError bound

1 nn aRerror

. The error is less than or equal to the first non-included term of the

series

You think…

When you see…

Use the Lagrange form of the remainder of a

Taylor polynomial centered at x=a.

Lagrange form of the remainder:Lagrange form of the remainder:

)(max!)1(

)( )1(

1

cfn

axxRerror n

n

n

Where c is some value between a and x.

You think…

When you see…

Find the velocity and acceleration vector of a particle at t=3 when

given x(t) and y(t)

Velocity and Acceleration VectorVelocity and Acceleration Vector

Velocity vector at t = 3:

(x’(3), y’(3))

Acceleration Vector at t = 3:

(x”(3), y”(3))

You think…

When you see…

Find the speed of a particle in motion on a

plane.

SpeedSpeed

22 ))(())(( tytx

Substitute in the appropriate value of t.

You think…

When you see…

Find the total distance of a particle moving in a plane on the interval

(t1, t2).

Total Distance TraveledTotal Distance Traveled

dttytxt

t 2

1

22 ))(())((

.

You think…

When you see…

Find the area of a polar graph between

21 and

Area of a Polar GraphArea of a Polar Graph

2

1

2

2

1

drA